Unit Circle: How to Memorize & Use

What is the Unit Circle?

The unit circle is a circle, centered at the origin, with a radius of 1.

This circle helps us find the exact values of some trigonometric functions and not the decimal approximations the calculator will give us. For example, it’ll help us find the exact value of sin or cos30°.

You don’t have to memorize all the values of the trigonometric functions, you just have to understand the unit circle.

Trigonometry of the Right Triangle and the Unit Circle

Let (x,y) be the point on the circle that is in the first quadrant.

The lengths x and y become the legs of a right triangle whose hypotenuse is actually the radius of our unit circle, i.e. the hypotenuse is 1.

Now let’s use some right angle trigonometry. Note that angle θ is acute.

This shows us that in a unit circle, cos θ = x and sin θ = y, which creates:

Sine is represented by the vertical leg.

Cosine is represented by the horizontal leg.

This formula applies to all the quadrants (it’s not limited to acute angles):

cos θ is the x coordinate of a point where terminal side of the angle intersects the unit circle

sin θ is the y coordinate of a point where terminal side of the angle intersects the unit circle

tan θ =

Using Pythagorean theorem, we have:

But since we have that x = cos θ and y = sin θ, this identity becomes:

This is called the Pythagorean trigonometric identity and it is very useful.

Trigonometric Functions

If you are asked to find sin60°, you would just need to look up for the y coordinate of the intersecting point on the circle. To find the cosine of the same angle, you just look up for the x coordinate of the same point.

There are some important angles whose sine, cosine and tangent you should memorize. They are given in the following table:

This might seem a lot at first, but there’s a trick to help you learn this faster.

Remembering the Unit Circle

For sine of 30°, 45°, and 60°, you should try to think “1, 2, 3” for the square root number in the numerator. The denominator is 2 and will stay the same.

sin 30° = or

sin 45° =

sin 60° =

For cosine of 30°, 45°, and 60°, you should try to think “3, 2, 1” for the square root number in the numerator. The denominator is 2 and will stay the same.

cos 30° =

cos 45° =

cos 60° = or

So actually, it’s just these three numbers: that you need to memorize.

Tangent will be equal to .

Another way to help you remember the 30° and 60° is the special triangle.

Sketch an equilateral triangle with side length 2. (All sides are 2 and all angles are 60°.)

Cut it in half. According to Pythagorean theorem, the new side is √3.

Now you can find the sine, cosine and tangent using soh cah toa. If you have trouble with this concept, please check out our in-depth SohCahToa guide.

Example: Unit Circle for θ in Quadrant I

Let’s do an example problem. What if you have θ=30º and you want to find the cosine and sine for it.

We can get the answer straight from the table, or we can use any of the tricks we have mentioned above. Let’s use the first one.

Remember that for sine of 30°, 45° and 60° we use “1, 2, 3” for the square root in the denominator, while the numerator stays 2. For cosine we use “3, 2, 1” and the rest is the same. Our angle is 30°

so we’ll take the first numbers:

Example: Unit Circle for θ in Quadrant II

What if you have θ=120º and you want to convert it to radians, identify the quadrant, and find its cosine and sine?

Step 1: To convert to radians, simply multiply with π/180

Step 2: This angle is in the second quadrant (greater than 90° and less than 180°). Using mnemostic ASTC (All Students Take Calculus), we can identify that only sine is positive in this quadrant. Cosine is negative here.

Step 3: When the angle is in Quadrant II, we will subtract it from 180 to figure the reference angle.

180°-120°=60°

Now we can solve for cosine and sine:

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